DIMENSIONAL REDUCTION IN ANYON SYSTEMS

Fractional statistics in one space dimension can be defined in two inequivalent ways: (i) By restricting the wave function for the relative two-body problem to the halfline, greater-than-or-equal-to 0, and imposing the boundary condition psi(x) = etapsi at x = 0. (ii) By quantizing the sp(1, R) algebra of observables x2 +/- p2 and xp + px, and noticing that the irreducible hermitian representations are labelled by a real parameter mu. We show that both these cases can be obtained by a dimensional reduction of a system of anyons in two dimensions. Case one corresponds to restricting the motion of the anyons to a line by a confining potential, and we give eta as a function of the statistics parameter theta for two different potentials. The second case corresponds to anyons in a magnetic field restricted to the first Landau level, and we find a linear relationship between mu and theta. We also construct coherent states corresponding to anyons in the lowest Landau level, and calculate the corresponding Berry connection. The statistics phase theta is shown to equal the Berry phase corresponding to an interchange of two anyons, thus generalizing previous results for bosons and fermions.